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Dimensional Analysis and Vectors

The idea on which dimensional analysis based is very simple and can be understood by all: physical laws do not depend on arbitrariness in the choice of basic units of measurement. An important conclusion can be drawn from this simple idea using simple arguments: the functions that express physical laws must posses a certain fundamental property, which in mathematics is called generalized homogeneity or symmetry. This property allows the number of arguments in these functions to be reduced, thereby making its simpler to obtain them. This is, in fact, the entire content of dimensional analysis - there is nothing more to it. Nevertheless, using dimensional analysis, researchers have been able to obtain remarkably deep results that have sometimes changed entire branches of science. The mathematical techniques required to derive these result also turned out to be simple and accessible to all.  

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Units are the chosen standards which when applied. In front of a number, gives complete information regarding the physical quantity. When a particular instrument indicates a reading, to specify the reading and use it in the further calculations, it is necessary to specify type and magnitude for that reading. The magnitude is nothing but the reading obtained on the instrument. The type of the reading is nothing but the unit of the physical quantity which is measured by the instrument. Without unit, only magnitude has no physical meaning. Thus the unit can be defined as follows: The standard measure of each type of physical quantity to be measured is called unit. Units are of two types: a) fundamental units b) derived units

Fundamental units: Fundamental units are those units which are independent and not related to each other. They have following properties:

a) They are well defined and of a suitable size.
b) They are easily reproducible at all places.
c) They do not vary with time, temperature, place and pressure etc.
d) They should be easily accessible.

Derived Units: The units which are derived from fundamental units are called derived units, e.g, units of area, volume, density, etc. 


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Regardless of the units employed a velocity is always a length divided by a time and a force is always a mass multiplied by a length and divided by time squared as seen from F = ma or F = $\frac{mv^{2}}{r}$. We write
[Force] = mass×length/time2 or
            = mass×length×time-2

The multiplying quantities (mass, length and time here) are the 'dimensions' of the derived quantity (force in the example used here). So the dimensions of a quantity are the base quantities from which it is made up in the same way that the dimensions of a box would be length, width and depth of the box. Square brackets are used to indicate 'the dimensions of' and the symbols M, L and T are used to denote mass, length and time when we are dealing with dimensions. Thus the dimensions of a force are M,L and T-2 are we can write
[F] = MLT-2

Some quantities are dimensionless, that is, their dimensions are zero. They are simply numbers, perhaps ratios of similar quantities. An angle is an example of dimensionless quantity. 

Dimensional Formula

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Units of a physical quantity expressed in terms of M, L and T is called its dimensional formula. 

Dimensional formulae of some important physical quantities

Physical Quantity  Dimensional Formula 
 Area                              [M0L2T0]                             
 Volume  [M0L3T0]
 Density  [ML-3T0]
 Velocity  [M0LT-1]
 Acceleration  [M0LT-2]
 Force  [MLT-2]
 Impulse  [MLT-1]
 Work  [ML2T-2]
 Power  [ML2T-3]
 Pressure  [ML-1T-2]

Application of Dimensional Equations

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Dimensional equation is the equation which equating the physical quantity with its dimensional formula. 

The unique quantity of every quantity which distinguishes it from all other quantities is called dimension. The dimensional symbols for fundamental units of length, mass and time are L, M and T respectively. The various powers of the fundamental units represent the dimensions of any derived unit. For example, the dimensional symbols for the derived units of volume, speed and acceleration are [L3], [LT-1] and [LT-2] respectively. These dimensional formulae of the derived units are very useful for converting units from one system to another. Some derived units, for convenience, have been given new names such as the derived unit of force in the S.I system is called the newton (N) instead of dimensionally correct name kgm/s2. All other mechanical quantities can be expressed in terms of three fundamental quantities, that is, length (L), mass (M) and time (T); a few of them are given below:

Force: mass×acceleration; [F] = [M][LT-2] = [MLT-2]
Work: force×distance; [W] = [MLT-2][L] = [ML2T-2]
Power: Work/Time; [P] = [ML2T-2]/[T] = [ML2T-3]

Such expressions indicate the nature of derived quantities in terms of fundamental dimensions [L], [M] and [T] and are known as dimensional equations. 

The dimensional equations help in serving the following purposes:
  • Conversion from one system of units to another one
  • Derivation of equations for physical quantities
  • Checking the accuracy of an equation

Vectors & their Representation

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The basic knowledge of scalars, vectors and their mathematical representation is important for understanding the phenomena associated with electric and magnetic fields. By definition, a scalar quantity sc represents just a magnitude, while a vector quantity represents both magnitude and direction. For example, mass of an object, temperature or electric charge distribution in a given region, time elapsed for a certain object to move, can be mathematically represented in terms of a scalar quantity. Similarly, force acting on an object, electric field or magnetic field distribution in a given region, electric current flowing along a conducting wire, can be mathematically represented in terms of a vector quantity. In general, the scalar and the vector quantities can be functions of one or more dependent physical parameters including spatial coordinate variables. The spatial dependence of a given problem or the three dimensional problem associated with the electric field and the magnetic field, can complicate the scalar quantity or the vector quantity representation. Sometimes, the graphical representation of the scalar quantity or the vector quantity helps to understand their physical significance. In fact, the scalar quantity can be graphically represented by a straight line of certain length with a specified scale factor assumed. The magnitude of the vector quantity can be graphically represented in the same manner, in addition, a straight bar or an arrow is included to depict the actual direction of the vector. The given figure, two arbitrary vectors $\vec{A}$ and $\vec{B}$ are represented graphically. 


Resolution of Vectors in to Rectangular Components

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Let us consider a vector $\vec{A}$ lying in the xy plane (figure) with its end at the origin of the coordinate system. Vector $\vec{A}$ can be expressed as the sum of vectors $\vec{ON}$ and $\vec{NP}$.

Resolution of Vectors
$\vec{OP}$ = $\vec{ON}$ + $\vec{NP}$
                 = A+ Ay
  $\vec{A}$ = $\hat{i}A_{x}$+$\hat{j}A_{y}$

Where Ax, Ay are the real numbers representing the length $\vec{ON}$ and $\vec{OM}$ (= $\vec{NP}$) respectively. Ax (= $\hat{i}A_{x}$) + Ay (= $\hat{i}A_{y}$) are the resolved rectangular components of the given vector  $\vec{A}$. Here Ax, Ay have unique values. We note that the lengths Ax, Ay are the projections of vectors  $\vec{A}$ on x and y axis respectively. If vector  $\vec{A}$ makes angle θ with x-axis,
Ax = A cos θ
Ay = A sin θ

Squaring the above equations and adding
A2cos2 θ + A2sinθ = AxAy2
or, A = $\sqrt{A_{x}^{2}+A_{y}^{2}}$

The equation relates the magnitude of vector A with the magnitudes of its resolved components. The direction of A with respect to Ax can also be expressed by the relationship

tan$\theta$ = $\frac{\vec{NP}}{\vec{ON}}$ (from $\bigtriangleup$ PON)

 $\theta$ = $\tan^{-1}$$\frac{A_{y}}{A_{x}}$

Addition and Subtraction of Vectors

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Vector Addition:
The resultant or vector sum of two displacement vectors is the displacement vector that results from performing first one and then the other displacement, this process is known as vector addition. However, the principle of addition has physical meaning for vector quantities other than displacements; for example, if two forces act on the same body then the resultant force acting on the body is the vector sum of the two. The addition of vectors only makes physical sense if they are of a like kind, for example if they are both forces acting in three dimensions. Vector addition is commutative. The generalization of this procedure to the addition of three (or more) vectors is clear and leads to the associative property of addition. Thus, it is immaterial in what order any number of vectors are added.

Vector Addition
Vector Subtraction:
The subtraction of two vectors is very similar to their addition. The subtraction of two equal vectors yields the zero vector, which has zero magnitude and no associated direction. 

Vector Subtraction
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